Timeline of computational mathematics

This is a timeline of key developments in computational mathematics.

18th century

Anton Felkel and Carl Hindenburg (independently of Felkel) build and use mechanical prime sieves based on the stencil method. Untitled, uncredited preamble to the "D.H. Lehmer's Number Sieves" article, on the preceding page

1890s

F. W. Lawrence provides a high-level design for a mechanical, automated prime sieve in his paper "Factorisation of numbers". (Thesis supervised by Hugh C. Williams.)

1910s

Lawrence's "Factorisation of numbers" is republished in French, French translation of sparking efforts to build a mechanical prime sieve. Eugène-Olivier Carissan builds his machine à congruences, a working mechanical sieve, in 1919.

1920s

D. H. Lehmer builds an electromechanical number sieve, the "bicycle" Lehmer sieve, in 1926

1930s

D. H. Lehmer builds further Lehmer sieves: the "gear sieve" in 1932 and the "film strip sieve" in 1936

1940s Monte Carlo simulation (voted one of the top 10 algorithms of the 20th century) invented at Los Alamos by von Neumann, Ulam and Metropolis.S. Ulam, R. D. Richtmyer, and J. von Neumann(1947). Statistical methods in neutron diffusion. Los Alamos Scientific Laboratory report LAMS–551.N. Metropolis and S. Ulam (1949). The Monte Carlo method. Journal of the American Statistical Association 44:335–341. Dantzig introduces the simplex algorithm (voted one of the top 10 algorithms of the 20th century). Systems Optimization Laboratory, Stanford University Huang Engineering Center (site host/mirror). First hydro simulations at Los Alamos occurred.Richtmyer, R. D. (1948). Proposed Numerical Method for Calculation of Shocks. Los Alamos, NM: Los Alamos Scientific Laboratory LA-671.A Method for the Numerical Calculation of Hydrodynamic Shocks. Von Neumann, J.; Richtmyer, R. D. Journal of Applied Physics, Vol. 21, pp. 232–237 Ulam and von Neumann introduce the notion of cellular automata.Von Neumann, J., Theory of Self-Reproducing Automata, Univ. of Illinois Press, Urbana, 1966. A routine for the Manchester Baby written to factor a large number (2^18), one of the first in computational number theory.The Manchester Mark 1. The Manchester group would make several other breakthroughs in this area.One tonne 'Baby' marks its birth: Dashing times. By Jonathan Fildes, Science and technology reporter, BBC News. LU decomposition technique first discovered.

1950s Hestenes, Stiefel, and Lanczos, all from the Institute for Numerical Analysis at the National Bureau of Standards, initiate the development of Krylov subspace iteration methods.Magnus R. Hestenes and Eduard Stiefel, Methods of Conjugate Gradients for Solving Linear Systems, J. Res. Natl. Bur. Stand. 49, 409–436 (1952).Eduard Stiefel, U¨ ber einige Methoden der Relaxationsrechnung (in German), Z. Angew. Math. Phys. 3, 1–33 (1952).Cornelius Lanczos, Solution of Systems of Linear Equations by Minimized Iterations, J. Res. Natl. Bur. Stand. 49, 33–53 (1952).Cornelius Lanczos, An Iteration Method for the Solution of the Eigenvalue Problem of Linear Differential and Integral Operators, J. Res. Natl. Bur. Stand. 45, 255–282 (1950). Voted one of the top 10 algorithms of the 20th century. Equations of State Calculations by Fast Computing Machines introduces the Metropolis–Hastings algorithm. Also, important earlier independent work by Alder and S. Frankel.Unfortunately, Alder's thesis advisor was unimpressed, so Alder and Frankel delayed publication of their results until much later. Alder, B. J., Frankel, S. P., and Lewinson, B. A., J. Chem. Phys., 23, 3 (1955).Stanley P. Frankel, Unrecognized Genius, HP9825.COM (accessed 29 Aug 2015). Enrico Fermi, Stanislaw Ulam, John Pasta, and Mary Tsingou, discover the Fermi–Pasta–Ulam–Tsingou problem.Fermi, E. (posthumously); Pasta, J.; Ulam, S. (1955) : Studies of Nonlinear Problems (accessed 25 Sep 2012). Los Alamos Laboratory Document LA-1940. Also appeared in 'Collected Works of Enrico Fermi', E. Segre ed., University of Chicago Press, Vol. II, 978–988, 1965. Recovered 21 Dec 2012 In network theory, Ford & Fulkerson compute a solution to the maximum flow problem.Ford, L. R.; Fulkerson, D. R. (1956). "Maximal flow through a network" . Canadian Journal of Mathematics. 8: 399–404. Householder invents his eponymous matrices and transformation method (voted one of the top 10 algorithms of the 20th century). Molecular dynamics invented by Alder and WainwrightAlder, B. J.; T. E. Wainwright (1959). "Studies in Molecular Dynamics. I. General Method". J. Chem. Phys. 31 (2): 459. Bibcode 1959JChPh..31..459A. doi:10.1063/1.1730376 John G.F. FrancisJ. G. F. Francis, "The QR Transformation, I", The Computer Journal, vol. 4, no. 3, pages 265–271 (1961, received Oct 1959) online at oxfordjournals.org; J. G. F. Francis, "The QR Transformation, II" The Computer Journal, vol. 4, no. 4, pages 332–345 (1962) online at oxfordjournals.org. and Vera KublanovskayaVera N. Kublanovskaya (1961), "On some algorithms for the solution of the complete eigenvalue problem," USSR Computational Mathematics and Mathematical Physics, 1(3), pages 637–657 (1963, received Feb 1961). Also published in: Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki [Journal of Computational Mathematics and Mathematical Physics], 1(4), pages 555–570 (1961). invent QR factorization (voted one of the top 10 algorithms of the 20th century).

1960s First recorded use of the term "finite element method" by Ray Clough,RW Clough, “The Finite Element Method in Plane Stress Analysis,” Proceedings of 2nd ASCE Conference on Electronic Computation, Pittsburgh, PA, Sept. 8, 9, 1960. to describe the methods of Courant, Hrenikoff and Zienkiewicz, among others. See also here. Using computational investigations of the 3-body problem, Minovitch formulates the gravity assist method.Minovitch, Michael: "A method for determining interplanetary free-fall reconnaissance trajectories," Jet Propulsion Laboratory Technical Memo TM-312-130, pages 38-44 (23 August 1961).Christopher Riley and Dallas Campbell, Oct 22, 2012. "The maths that made Voyager possible" . BBC News Science and Environment. Recovered 16 Jun 2013. Molecular dynamics was invented independently by Aneesur Rahman. Cooley and Tukey re-invent the Fast Fourier transform (voted one of the top 10 algorithms of the 20th century), an algorithm first discovered by Gauss. Edward Lorenz discovers the butterfly effect on a computer, attracting interest in chaos theory. Kruskal and Zabusky follow up the Fermi–Pasta–Ulam–Tsingou problem with further numerical experiments, and coin the term "soliton".Zabusky, N. J.; Kruskal, M. D. (1965). "Interaction of 'solitons' in a collisionless plasma and the recurrence of initial states". Phys. Rev. Lett. 15 (6): 240–243. Bibcode 1965PhRvL..15..240Z. doi:10.1103/PhysRevLett.15.240.http://www.merriam-webster.com/dictionary/soliton; retrieved 3 nov 2012. Birch and Swinnerton-Dyer conjecture formulated through investigations on a computer.Birch, Bryan; Swinnerton-Dyer, Peter (1965). "Notes on Elliptic Curves (II)". J. Reine Angew. Math. 165 (218): 79–108. doi:10.1515/crll.1965.218.79. Grobner bases and Buchberger's algorithm invented for algebraBruno Buchberger: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal (PDF; 1,8 MB). 1965 Frenchman Verlet (re)discovers a numerical integration algorithm, (first used in 1791 by Delambre, by Cowell and Crommelin in 1909, and by Carl Fredrik Störmer in 1907, hence the alternative names Störmer's method or the Verlet-Störmer method) for dynamics. Risch invents algorithm for symbolic integration.Risch, R. H. (1969). "The problem of integration in finite terms". Transactions of the American Mathematical Society. American Mathematical Society. 139: 167–189. doi:10.2307/1995313. JSTOR 1995313. Risch, R. H. (1970). "The solution of the problem of integration in finite terms". Bulletin of the American Mathematical Society. 76 (3): 605–608. doi:10.1090/S0002-9904-1970-12454-5.

1970s Mandelbrot, from studies of the Fatou, Julia and Mandelbrot sets, coined and popularized the term 'fractal' to describe these structures' self-similarity.B. Mandelbrot; Les objets fractals, forme, hasard et dimension (in French). Publisher: Flammarion (1975), ; English translation Fractals: Form, Chance and Dimension. Publisher: Freeman, W. H & Company. (1977). .Mandelbrot, Benoît B.; (1983). The Fractal Geometry of Nature. San Francisco: W.H. Freeman. . Kenneth Appel and Wolfgang Haken prove the four colour theorem, the first theorem to be proved by computer.Kenneth Appel and Wolfgang Haken, "Every planar map is four colorable, Part I: Discharging," Illinois Journal of Mathematics 21: 429–490, 1977.Appel, K. and Haken, W. "Every Planar Map is Four-Colorable, II: Reducibility." Illinois J. Math. 21, 491–567, 1977.Appel, K. and Haken, W. "The Solution of the Four-Color Map Problem." Sci. Amer. 237, 108–121, 1977.

1980s Fast multipole method invented by Rokhlin and Greengard (voted one of the top 10 algorithms of the 20th century).L. Greengard, The Rapid Evaluation of Potential Fields in Particle Systems, MIT, Cambridge, (1987).Rokhlin, Vladimir (1985). "Rapid Solution of Integral Equations of Classic Potential Theory." J. Computational Physics Vol. 60, pp. 187–207.L. Greengard and V. Rokhlin, "A fast algorithm for particle simulations," J. Comput. Phys., 73 (1987), no. 2, pp. 325–348.

1990s The appearance of the first research grids using volunteer computing – GIMPS (1996) and distributed.net (1997). Kepler conjecture is almost all but certainly proved algorithmically by Thomas Hales in 1998.

2000s In computational group theory, God's Number for the Rubik's Cube is shown to be 20.The Rubik's Cube Conjecture PROVEN! (Do we care?) Wednesday, September 08, 2010God's Number is 20. Mathematicians completely map the E8-group.Math research team maps E8: Calculation on paper would cover Manhattan. MIT News. Elizabeth A. Thomson, News Office; March 18, 2007.E8 Media Blitz, Peter Woit.Mathematicians Map E8. By Armine Hareyan 2007-03-20 02:21.

2010s Hales completes the proof of Kepler's conjecture.What is the way of packing oranges? — Kepler's conjecture on the packing of spheres. Posted on May 26, 2015 by Antoine Nectoux. Klein Project Blog: Connecting mathematical worlds.Announcement of Completion. Flyspeck Project, Google Code.Proof confirmed of 400-year-old fruit-stacking problem. New Scientist, 12 August 2014.

Timeline of computational mathematics

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